This book is intended for researchers and students concerned with questions in analysis and function theory. The author provides an exposition of the main results obtained in recent years by Soviet and other mathematicians in the theory of mappings with bounded distortion, an active direction in contemporary mathematics. The mathematical tools presented can be applied to a broad spectrum of problems that go beyond the context of the main topic of investigation. For a number of questions in the theory of partial differential equations and the theory of functions with generalized derivatives, this is the first time they have appeared in an internationally distributed monograph.

Table of Contents

Introduction

Some facts from the theory of functions of a real variable

Functions with generalized derivatives

Möbius transformations

Definition of a mapping with bounded distortion

Mappings with bounded distortion on Riemannian spaces

Main facts in the theory of mappings with bounded distortion

Estimates of the moduli of continuity and differentiability almost everywhere of mappings with bounded distortion

Some facts about continuous mappings on \(R^n\)

Conformal capacity

The concept of the generalized differential of an exterior form

Mappings with bounded distortion and elliptic differential equations

Topological properties of mappings with bounded distortion

Local structure of mappings with bounded distortion

Characterization of mappings with bounded distortion by the property of quasiconformity

Sequences of mappings with bounded distortion

The set of branch points of a mapping with bounded distortion and locally homeomorphic mappings

Extremal properties of mappings with bounded distortion

Some further results

Some results in the theory of functions of a real variable and the theory of partial differential equations

Functions with bounded mean oscillation

Harnack's inequality for quasilinear elliptic equations

Theorems on semicontinuity and convergence with a functional for functionals of the calculus of variations